the relationship between basic arithmetic operations is illustrated, allowing the higher operations to be defined naturally as above. The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term;[14] so a is the base, b is the exponent (or hyperexponent),[12] and n is the rank (or grade).[6]

In common terms, the hyperoperations are ways of compounding numbers that increase in growth based on the iteration of the previous hyperoperation. The concepts of successor, addition, multiplication and exponentiation are all hyperoperations; the successor operation (producing x+1 from x) is the most primitive, the addition operator specifies the number of times 1 is to be added to itself to produce a final value, multiplication specifies the number of times a number is to be added to itself, and exponentiation refers to the number of times a number is to be multiplied by itself.

One of the earliest discussions of hyperoperations was that of Albert Bennett[6] in 1914, who developed some of the theory of commutative hyperoperations (see below). About 12 years later, Wilhelm Ackermann defined the function [15] which somewhat resembles the hyperoperation sequence.

In his 1947 paper,[5]R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, etc.). As a three-argument function, e.g., , the hyperoperation sequence as a whole is seen to be a version of the original Ackermann function — recursive but not primitive recursive — as modified by Goodstein to incorporate the primitive successor function together with the other three basic operations of arithmetic (addition, multiplication, exponentiation), and to make a more seamless extension of these beyond exponentiation.

The original three-argument Ackermann function uses the same recursion rule as does Goodstein's version of it (i.e., the hyperoperation sequence), but differs from it in two ways. First, defines a sequence of operations starting from addition (n = 0) rather than the successor function, then multiplication (n = 1), exponentiation (n = 2), etc. Secondly, the initial conditions for result in , thus differing from the hyperoperations beyond exponentiation.[7][16][17] The significance of the b + 1 in the previous expression is that = , where b counts the number of operators (exponentiations), rather than counting the number of operands ("a"s) as does the b in , and so on for the higher-level operations. (See the Ackermann function article for details.)

In 1928, Wilhelm Ackermann defined a 3-argument function which gradually evolved into a 2-argument function known as the Ackermann function. The original Ackermann function was less similar to modern hyperoperations, because his initial conditions start with for all n > 2. Also he assigned addition to n = 0, multiplication to n = 1 and exponentiation to n = 2, so the initial conditions produce very different operations for tetration and beyond.

n

Operation

Comment

0

1

2

3

An offset form of tetration. The iteration of this operation is different than the iteration of tetration.

In 1984, C. W. Clenshaw and F. W. J. Olver began the discussion of using hyperoperations to prevent computer floating-point overflows.[23] Since then, many other authors[24][25][26] have renewed interest in the application of hyperoperations to floating-point representation. (Since Hn(a, b) are all defined for b = -1) While discussing tetration, Clenshaw et al. assumed the initial condition , which makes yet another hyperoperation hierarchy. Just like in the previous variant, the fourth operation is very similar to tetration, but offset by one.

Commutative hyperoperations were considered by Albert Bennett as early as 1914,[6] which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule

which is symmetric in a and b, meaning all hyperoperations are commutative. This sequence does not contain exponentiation, and so does not form a hyperoperation hierarchy.

^ abcda [n] (-1) = 0 for all real number a and all integer n ≥ 4, because if so, than a [n] 0 = a [n - 1] (a [n] (-1)) = a [n - 1] 0 = 1 (because of n ≥ 4), this is keeping with the define: a [n] 0 = 1 for all real number a.